ℐ-αβ-statistical relative uniform convergence for double sequences and its applications

Generalized equi-statistical convergence of positive linear operators and associated approximation theorems

Statistical convergence was extended to weighted statistical convergence in [24], by using a sequence of real numbers sk, satisfying some conditions. Later, weighted statistical convergence was considered in [35] and [19] with modified conditions on sk. Weighted statistical convergence is an extension of statistical convergence in the sense that, for sk = 1, for all k, it reduces to statistical convergence. A definition of weighted ??-statistical convergence of order ?, considered in [25] does not have this property. To remove this extension problem the definition given in [25] needs some modifications. In this paper, we introduced the modified version of weighted ??-statistical convergence of order ?, which is an extension of ??-statistical convergence of order ?. Our definition, with sk = 1, for all k, reduces to ??-statistical convergence of order ?. Moreover, we use this definition of weighted ??-statistical convergence of order ?, to prove Korovkin type approximation theorems via, weighted ??-equistatistical convergence of order ? and weighted ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Also we prove Korovkin type approximation theorems via ??-equistatistical convergence of order ? and ??-statistical uniform convergence of order ?, for bivariate functions on [0,?) x [0,?). Some examples of positive linear operators are constructed to show that, our approximation results works, but its classical and statistical cases do not work. Finally, rates of weighted ??-equistatistical convergence of order ? is introduced and discussed.

The idea of statistical convergence was given by Zygmund [45] in the first edition of his monograph published in Warsaw in 1935 and then was considered as a summability method by Schoenberg [38] in 1959 despite the fact that it made its initial appearance in a short note by Fast [16] and Steinhaus [42] in 1951. Along with the theory of summability it has played an important role in Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Salat [35] also showed that the set of bounded statistical convergent real valued sequences is a closed subspace of bounded sequences in 1980. After that Fridy [17] introduced the concept of statistically Cauchiness of sequences and proved that it is equivalent to statistical convergence. More importantly he proved that there is no any matrix summability method which involves statistical convergence in the same paper in 1985. Another magnificent development was presented to the literature by Connor [8] in 1988. For the first time in the literature his work confirmed the direct link between statistical convergence and strong \(s-\)Cesàro summability by unveiling that the notions are equivalent for bounded sequences, where \(0<s<\infty \). Beside he showed that the set of statistically convergent sequences does not generate a locally convex \(FK-\)space. Recently, generalizations of statistical convergence have started to arise in many articles by several authors. Mursaleen [32] gave the concept of \(\lambda -\)statistical convergence in 2000, while Savaş [36] examined the relationship between strong almost convergence and almost \(\lambda -\)statistical convergence in the same year. Afterward, in 2010, Colak [5] made a new approach to the concept by studying the notion of statistical convergence of order \(\alpha \) where \(\alpha \in (0,1]\). Nowadays statistical convergence has been studied by many mathematicians such as Bhardwaj et al. [3, 4], Colak and Bektas [7] , Connor [8, 9], Et et al. [10,11,12,13,14, 37], Fridy and Orhan [18,19,20], Gadjiev and Orhan [21], Hazarika et al. [22,23,24], Isik et al. [25,26,27], Kolk [28], Mohiuddine et al. [30, 31], Rath and Tripathy [34], Sengul et al. [39,40,41] and many others. In 1932 Agnew [1] presented deferred Cesàro mean by modifying Cesàro mean to obtain more useful methods including stronger features which do not belong to nearly all methods. Kucukaslan and Yılmaztürk [29] came up with the idea of combining the deferred Cesàro mean and the concept of statistical convergence. This gave them the opportunity to generalize both strong \(s-\)Cesàro summability and statistical convergence with the sense of deferred Cesàro mean. In this study, we introduce the concepts of deferred statistical convergence of order \(\alpha \beta \) and strongly \(s-\)deferred Cesàro summability of order \(\alpha \beta \) of complex (or real) sequences.KeywordsStatistical convergenceCesàro summabilityDeferred Cesàro meansDeferred statistical convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.

The concept of statistical convergence in 2-normed spaces for double sequence was introduced in [17].In the rst , we introduce concept strongly statistical convergence in 2- normed spaces and generalize some results. moreover, we dene the concept of statistical uniform convergence in 2

Soft sets and statistical convergence are both mathematical tools with which some generalizations can be made. In this study, we defined the weighted statistical convergence of soft point sequences in soft topological spaces and examined it from some aspects. Furthermore, statistical convergence was attained using 1-density, a more versatile density function than asymptotic density. Regarding weighted soft statistical convergence concept, we also establish some relationships between soft topology and the classical topology induced by it within the framework of the statistical convergence concept and some clusters associated with statistical convergence were defined.

On lacunary statistical convergence with respect to the intuitionistic fuzzy normed space

Tauberian theorems for statistically convergent double sequences

The notion of deferred weighted statistical probability convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the deferred weighted statistical convergence. Such concepts were introduced and studied by Srivastava et al. (Appl Anal Discrete Math, 2020). In the present work, we introduced and studied the notion of statistical probability convergence as well as statistical convergence for sequences of random variables and sequences of real numbers respectively defined over a Banach space via deferred Norlund summability mean. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space and demonstrated that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). Finally, an illustrative example is presented here by the generalized Meyer-Konig and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

In this paper, we introduce the concepts of statistical monotone convergence and statistical order convergence in a Riesz space, and establish some basic facts. We show that the statistical order convergence and the statistical convergence in norm need not be equivalent in a normed Riesz space. Finally, we introduce the statistical order boundedness of a sequence in a Riesz space.

Statistical convergence has recently attracted the wide-spread attention of researchers due mainly to the fact that it is more general than the classical convergence. Furthermore, the notion of equi-statistical convergence is stronger than that of the statistical uniform convergence. Such concepts were introduced and studied by Balcerzak et al. (J Math Anal Appl 328:715–729, 2007). In this paper, we have used the notion of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence in conjunction with the deferred Norlund statistical convergence in order to establish several inclusion relations between them. We have also applied our presumably new concept of the deferred Norlund equi-statistical convergence to prove a Korovkin type approximation theorem and demonstrated that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems which were proven by earlier authors. Finally, we consider a number of interesting cases in support of our definitions and results presented in this paper.

On the ideal convergence of double sequences in intuitionistic fuzzy normed spaces

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the concept of statistical convergence for sequences of real numbers, which are defined over a Banach space via deferred weighted summability mean. We first establish a theorem presenting a connection between them. Based upon our proposed methods, we then prove a new Korovkin-type approximation theorem with periodic test functions for a sequence of random variables on a Banach space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in statistical versions). We also estimate the rate of deferred weighted statistical probability convergence and accordingly establish a new result. Finally, an illustrative example is presented here by means of the generalized Fej?r convolution operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

The statistical convergence for sequences of fuzzy-number-valued functions

The concepts of statistical convergence and strong p-Cesaro summability of sequences of real numbers were introduced in literature independently, and it was shown that if a sequence is strongly p-Cesaro summable, then it is statistically convergent and also a bounded statistically convergent sequence must be p-Cesaro summable. In the present paper, two new concepts named statistical convergence of order $$\left( \beta ,\gamma \right) $$ and strongly p-Cesaro summability of order $$\left( \beta ,\gamma \right) $$ are introduced for sequences of fuzzy numbers, where $$\alpha $$ and $$\beta $$ real numbers such that $$0<\alpha \le \beta \le 1$$ and some relations between statistical convergence of order $$\left( \beta ,\gamma \right) $$ and strongly p-Cesaro summability of order $$\left( \beta ,\gamma \right) $$ are given. Furthermore, it is shown that a bounded and statistically convergent sequence of fuzzy numbers need not strongly p-Cesaro summable of order $$\left( \beta ,\gamma \right) $$ in general for $$0<\beta \le \gamma \le 1$$ .